The use of coded waveforms in space-variant optical signal processors using coordinate transformations is considered. It is shown that nonlinear transmitted coded signals must be used with such a processor and that this results in novel waveform design and system approaches for radar and communications

Casasent, David

Psaltis, Demetri

English

Caltech Authors - Main

18 OPTICS LETTERS / Vol. 4, No. I / January 1979
Waveform considerations in space-variant optical processors
David Casasent and Demetri Psaltis
Department of Electrical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
Received August 24, 1978
The use of coded waveforms in space-variant optical signal processors using coordinate transformations is consid-
ered. It is shown that nonlinear transmitted coded signals must be used with such a processor and that this results
in novel waveform design and system approaches for radar and communications.
Space-variant optical processors are the subject of
considerable research.1 One of the most promising
methods by which such systems can be realized is the
application of coordinate transformations to the input
data.2' 3 The extension of this technique to produce
optical Mellin transforms4 and distortion-invariant
pattern-recognition systems has been reported over the
past three years.k7 Of particular interest in this Letter
is the use of the Mellin transform in Doppler-invariant
signal processors, as we described earlier.8' 9 However,
because the Mellin transform is a space-variant opera-
tion, a time-sequential implementation of the required
coordinate transformation would be required unless a
fully parallel, real-time, coordinate transformation
system can be developed. With such a properly con-
figured system, a single-channel 1-D optical processor
can be used to realize the necessary range/Doppler
processing.
At least two techniques have been reported by which
the required parallel coordinate transformation can be
realized. One scheme that is appropriate for 2-D inputs
utilizes computer-generated holograms1 0 -12 but suffers
from input space-bandwidth product limitations. A
second method13'14 is of use with 1-D input functions
and 1-D distortions only and results in large light losses.
However, with refinements in these methods, our sin-
gle-channel 1-D, real-time Mellin-transform Doppler-
invariant-processor concept8' 9 seems more realizable.
Hence, in this Letter, we extend our earlier work8'9
to the case of coded waveforms, as are conventionally
used in radar and communications. We now describe
in more detail the waveform properties and operation
of such space-variant optical processors with complex
coded waveforms.
Following a review of Mellin-transform-based Dop-
pler-invariant signal processing we show that previously
described space-variant signal processors8' 9' 15 cannot
operate on conventional coded waveforms without a
severe loss in system performance. Rather, proper
operation of such processors requires transmission codes
that are nonlinear. Such waveforms are then shown to
result in a novel spread-spectrum coding that promises
to provide new research avenues for waveform design
in spread-spectrum, radar, and communications sys-
tems. As we have long advocated, optical signal pro-
cessors should not be limited by the conventional
waveform and system-design constraints introduced by
existing processing technologies. The case at hand
serves to demonstrate this point vividly. In future
papers, we will describe the ambiguity functions, sys-
tem-design parameters, and the advantageous noise,
jammer, and interference-rejection features of such
systems using nonlinear waveforms and space-variant
optical processors.
In the general formulation of space-variant pattern
recognition, we represent the undistorted input signal
(in 1-D for simplicity) by f(x) and a distorted version
of it by
f'(x) = f[g(x,a)], (1)
where g(x,a) is the distorting function and a an un-
known distortion parameter. To correlate f(x) and
f'(x) invariant to the distortion g(x,a), we apply the
coordinate transformation 7
t = h-l(x) = f _xO / dx
da -a9g/d~a (2)
to f (x) and f'(x), where to is a function of a alone. This
transformation produces two new functions, f1(t) andf1 '(t), which when correlated yield4 -9
R(t) = fl(t) (ifl'(t) = fl(t) ®fi(t) * b(i - to), (3)
where 0 denotes correlation, * represents convolution,
and t is the correlation plane variable. The unknown
distortion parameter a can be found from the location
= to) of the output correlation peak.
When the input distortion is g(x,a) = ax, f'(x) = f(ax)
is a Doppler-shifted version of f(x). Solving Eq. (2), we
find that t = -In x and to = In a. Thus the resultant
system of Fig. 1 with f1'(t) recorded at Po and Fi*(u) =
5([fI(t)] recorded at P1 becomes a space-variant Dop-
pler-invariant 1-D signal processor in which i corre-
Po L1 PI L2 P2
- 1X rq t j iDOPPLER
flx) )Ny1x)]* 1f1~f
Ft1u)
Fig. 1. Schematic diagram of a space-variant, Doppler-in-
variant optical signal processor.
0146-9592/79/010018-03$0.50/0 © 1978, Optical Society of America
January 1979 / Vol. 4, No. 1 / OPTICS LETTERS 19
TRANSMIT- LIN
AUTO CORREL
t.exp t * t
(a)
TRANSMIT - LIN
t. ept .tL
.200
DISCRETE JAMMER
AT _ i_50
(b)
Fig. 2. Cross-sectional scan of the optically produced cor-
relation of a nonlinear coordinate-transformed Barker-coded
waveform: (a) autocorrelation, (b) jammer.
sponds to the Doppler channel. With a time-varying
input function at Po, the time of occurrence of the P2
output peak denotes the target's range, whereas the
spatial position of the peak denotes its Doppler.
We now concern ourselves with the performance of
the system of Fig. 1 when f(t) is a coded waveform. For
the case of binary phase-modulated waveforms with
constant amplitude modulation, we represent the real
form of f(t) by
f(t) = cos[cot + 0 (t)] = a(t)cos(coot), (4)
where a(t) = exp[.j0 (t)], wo is the carrier, ( denotes the
phase modulation, and a (t) denotes the corresponding
bipolar binary amplitude modulation. For the case of
a 13-bit Barker biphase code,1 6
an(t) = expUj0,(t)]
= +1,+1,+1,+1,+1,-1,-1,+1,+1,-1,+1,-1,±1. (5)
For the remainder of this Letter, we consider only the
use of the x = ln t coordinate transformation (because
of the Doppler invariance that results) and the 13-bit
Barker coded waveform of Eqs. (4) and (5). No loss of
generality results from this choice. Further analysis is
merely simpler if a specific example is used.
To demonstrate the factors that affect the choice of
the coded waveform used, we apply t = exp x to Eq. (4),
and, with the variable change x = t, we find that
f(exp t) = fi(t) = cos[wo exp t + ( (exp t)]. (6)
In the conventional space-variant signal-processing
approach described above, we would correlate this fl(t)
reference signal with a similarly coordinate-transformed
received signalfl'(t). For the no-distortion case, fl(t)
= f1'(t), and the resultant autocorrelation is
R(t) = f(exp t) ® f(exp t). (7)
In Fig. 2(a) we show the cross-sectional scan of an
optically produced correlation in Eq. (7) of the nonlinear
13-bit Barker code described by Eq. (6) with bandwidth
= 11.4 cycles/mm. Various methods to realize the re-
quired coordinate transformation were noted above.
For the data in Fig. 2, a log module was used in the de-
flection system of a TV monitor. The measured width
of the correlation peak (including output display mag-
nification) is 1 mm. To compare this with the perfor-
mance possible with a linear or normal Barker code of
the same bandwidth (BW), we would use a bit length of
2/11.4 = 0.18 mm and expect an output-correlation peak
width or system resolution of 0.36 mm. Experimental
data on such a linear Barker code confirmed that such
resolution was obtainable on the same optical system.
One such typical output is shown in Fig. 2(b). Many
other analogous output correlations on various wave-
forms have been investigated and found to yield similar
results. Thus, as was just demonstrated, although the
correlation of a nonlinear Barker code as in Eq. (7) re-
sults in a good correlation peak, we could achieve nearly
three times the performance or resolution (i.e., 1
mm/0.36 mm = 3) with a normal Barker code of com-
parable BW.
A stationary phase analysis17 of Eq. (7) with typical
wo and 0 values will show that the nonlinear carrier
rather than the phase code determines the BW of fI(t)
and all characteristics of R (t). This is due to the much
larger phase change introduced by the nonlinear carrier
in Eq. (6). Because of the spread but highly tapered
spectrum of Eq. (6), the Doppler resolution AP of the
system is poor, and better resolution would result if a
normal code of the same increased bandwidth were
used. Of more concern is the fact that the waveform
and system designer now has no tools at his disposal,
since changing the phase code 0 in Eq. (6) does not ap-
preciably alter R (i) in Eq. (7). Similarly, when a range
i- shift is present, the correlation will not be appreciably
different, since the major contribution to R(t) is pro-
duced by the coordinate-transformed carrier. This
aspect is disastrous when a jammer is present, since it
will correlate equally well with fl(t) because both are
subjected to the same coordinate transformation. In
Fig. 2(b), we show the cross-sectional scan of the cor-
relation of a discrete jammer with the reference wave-
form in Eq. (6) to demonstrate this. For these and
many other reasons, such operation of the system and
the use of normal coded waveforms in this type of
space-variant processor will produce poor system per-
formance.
The above problems can be alleviated if we transmit
a waveform that, when coordinate transformed, results
in a normal Barker (or any other) coded signal. For the
Mellin-transform coordinate transformation under
consideration, the optimum transmitted signal would
thus be
ft(t) = coslwo ln( t + to) + 0 [ln(t + to)]) (8)
or a ln-scaled version of Eq. (4) with the coordinate
transformation
t = in(t + to) (9)
applied. The to variable in Eq. (9) is necessary since ln
0 = - . On reception of ft (t) in Eq. (8) or a Doppler-
or range-shifted version of it, we would apply the inverse
coordinate transformation
t = exp x - t, (10)
20 OPTICS LETTERS / Vol. 4, No. 1 / January 1979
f(x) x = In(t + t M) -'(-TRANSMIT
f(x)
_ _ _ _ _ _ _ _ _ _ _ f ,( t)
' = exp x - - -e RECEIVE
-X(tx) = X(ryv)
Fig. 3. Block diagram of space-variant optical-ambiguity-
function processor for use with coded waveforms.
X MSF i DOPPLER
RFF
-INPUT -
-LASER -. o o
- LIGHT P0 L- 12 t U
PO LI P, L2 P2 -RANGE
Il( if-tI) YP[IIJ R(t.1)
Fig. 4. Schematic diagram of a single-channel space-variant
optical processor to calculate the 2-D ambiguity function for
coded waveforms.
yielding a new function,
fti(t) = ft(ex - i) = f,(ix). (11)
The desirable range and Doppler information for this
system are obtained by correlating Eq. (11) with Eq. (4)
continuously for all i and x,
R(i,x) = f (i,x) Gf(x), (12)
where x is the Doppler or correlation variable and i is
the continuously varying time variable. A block di-
agram of the final system is shown in Fig. 3.
In this approach, f(x) is coordinate transformed,
transmitted, and, on receipt, inverse coordinate trans-
formed; this signal is then correlated with the original
f (x) waveform. This approach is far preferable to the
correlation of coordinate-transformed versions of the
transmitted arid received signals, as noted above.
A candidate optical system to realize the functional
diagram of Fig. 3 is shown in Fig. 4. One of the real-
time and parallel optical space-variant systems noted
above is used to realize the transformation from Eq. (8)
to Eq. (11). With the transmittance of PO given by Eq.
(11) and a matched spatial filter of Eq. (4) recorded at
P1 , the system of Fig. 3 realizes the correlation and
ambiguity-function generation described by Eq. (12).
In this system, the position in x at P2 at which the cor-
relation appears will correspond to the target's Doppler,
and the time t at which the correlation occurs will cor-
respond to the target's range.
The many desirable waveform-design and noise-
rejection features of this system will be described in
later papers. The transmitted waveform in Eq. (12) has
many unique features. It uses a nonlinear code, thus
decreasing its detectability by a third party. Since the
BW of the original signal is spread and spread nonlin-
early, a large noise immunity to jamming and interfer-
ence results for the same reason that spread-spectrum
systems enjoy such noise immunity. The nonlinear
coordinate transformation produces a further level of
noise and jammer rejection and an increased system-
and waveform-design flexibility. Finally, because the
carrier in Eq. (8) is nonlinear, its wo and pulse-repetition
frequency cannot be easily determined.
This work was supported by grant ENG77-20038
from the National Science Foundation.
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Waveform considerations in space-variant optical processors

The use of coded waveforms in space-variant optical signal processors using coordinate transformations is consid-ered. It is shown that nonlinear transmitted coded signals must be used with such a processor and that this results in novel waveform design and system approaches for radar and communications. Space-variant optical processors are the subject of considerable research.1 One of the most promising methods by which such systems can be realized is the application of coordinate transformations to the input data.2 &apos; 3 The extension of this technique to produce optical Mellin transforms4 and distortion-invariant pattern-recognition systems has been reported over the past three years.k7 Of particular interest in this Letter is the use of the Mellin transform in Doppler-invariant signal processors, as we described earlier.8 &apos; 9 However, because the Mellin transform is a space-variant opera-tion, a time-sequential implementation of the require

Waveform considerations in space-variant optical processors

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